The best way to see
10 persons and we wish to divide them at random into 3 groups consisting of 5, 3, and 2
is to think of 10 slots, there are 10! ways of placing 10 items in these 10 positions.
Then break those 10 into three groups, the first 5, the next 3 and the last 2.
You end up with "double counting" of 5! in the first 5 spots since any rearrangement of those 5 is still the same.
Likewise for the next two groups.
Hence the answer is